fSchur

The function B=fSchur(A,T_1 computes the Schur complement of the block structured symmetric matrix A with respect to the picking matrix T_1 as follows:

Let T_1 be a given picking matrix

    \[T_1=\left(\begin{array}{ccc}0&0&\cdots\\ \vdots& \vdots& \cdots\\ 0&\vdots&\cdots\\I&\vdots&\cdots\\0&0&\cdots\\ \vdots&I&\cdots\\\vdots&0&\cdots\\\vdots&\vdots&\cdots\\0&0&\cdots\end{array}\right)\]

and let T_2 be the complementary picking matrix (i.e., the orthogonal complement) such that T^TT=\left(T_1\ T_2\right)^T \left(T_1\ T_2\right)=I. For a given symmetric matrix A, the permutation operation T^TAT then yields the matrix

    \[T^TAT=\left(\begin{array}{cc}T_1^TAT_1& T_1^TAT_2\\T_2^TAT_1& T_2^TAT_2 \end{array}\right) =\left(\begin{array}{cc}Q&S\\S^T&R \end{array}\right).\]

Subsequently, the function computes the Schur complement as B=R-S^TQ^{-1}S.

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